AN INTRODUCTION TO THE EINSTEIN–VLASOV SYSTEM

BANACH CENTER PUBLICATIONS, VOLUME 41 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES

WARSZAWA 1997

AN INTRODUCTION TO THE EINSTEIN–VLASOV SYSTEM

A L A N D. R E N D A L L

Max-Planck-Institut f¨ur Gravitationsphysik Schlaatzweg 1, 14473 Potsdam, Germany

E-mail: rendall@aei-potsdam.mpg.de

1. Introduction. The Vlasov equation arises in kinetic theory. It gives a statistical description of a collection of particles. It is distinguished from other equations of kinetic theory by the fact that there is no direct interaction between particles. In particular, no collisions are included in the model. Each particle is acted on only by fields which are generated collectively by all particles together. The fields which are taken into account depend on the physical situation being modelled. In plasma physics, where this equation is very important, the interaction is electromagnetic and the fields are described either by the Maxwell equations or, in a quasi-static approximation, by the Poisson equation [26]. In gravitational physics, which is the subject of the following, the fields are described by the Einstein equations or, in the Newtonian approximation, by the Poisson equation. (There is a sign difference in the Poisson equation in comparison with the electromagnetic case due to the replacement of a repulsive by an attractive force.) The best known applications of the Vlasov equation to self-gravitating systems are to stellar dynamics [3]. It can also be applied to cosmology. In the first case the systems considered are galaxies or parts of galaxies where there is not too much dust or gas which would require a hydrodynamical treatment. Possible applications are to globular clusters, elliptical galaxies and the central bulge of spiral galaxies. The ‘particles’ in all these cases are stars. In the cosmological case they are galaxies or even clusters of galaxies. The fact that they are modelled as particles reflects the fact that their internal structure is believed to be irrelevant for the dynamics of the system as a whole.

These lectures are concerned not with the above physical applications but with some basic mathematical aspects of the Einstein–Vlasov system. First the definition and gen- eral mathematical properties of this system of partial differential equations are discussed and then the Cauchy problem for this system is formulated. The central theme in what follows is the global Cauchy problem, where ‘global’ means global in time. Up to now

1991 Mathematics Subject Classification: Primary 83C05; Secondary 83C55.

The paper is in final form and no version of it will be published elsewhere.

[35]

global results have only been obtained in very special cases and one of these, the case of spherically symmetric asymptotically flat solutions, is discussed in detail. The global existence results presented depend on having a suitable local existence result. At first the necessary local existence theorem is stated without proof and used in obtaining global theorems. The proof of the local existence theorem is discussed in some detail afterwards.

Further information on kinetic theory in general relativity may be found in [10].

Let (M, g_αβ) be a spacetime, i.e. M is a four-dimensional manifold and g_αβis a metric of Lorentz signature (−, +, +, +). Note that gαβdenotes a geometric object here and not the components of the geometric object in a particular coordinate system. In other words the indices are abstract indices. (See [28], section 2.4 for a discussion of this notation.) It is always assumed that the metric is time-orientable, i.e. that the two halves of the light cone at each point of M can be labelled past and future in a way which varies continuously from point to point. With this global direction of time, it is possible to distinguish between future-pointing and past-pointing timelike vectors. The worldline of a particle of non-zero rest mass m is a timelike curve in spacetime. The unit future-pointing tangent vector to this curve is the 4-velocity v^αof the particle. Its 4-momentum p^αis given by mv^α. There are different variants of the Vlasov equation depending on the assumptions made. Here it is assumed that all particles have the same mass m but it would also be possible to allow a continuous range of masses. When all the masses are equal, units can be chosen so that m = 1 and no distinction need be made between 4-velocity and 4-momentum.

There is also the possibility of considering massless particles, whose wordlines are null curves. In the case m = 1 the possible values of the four-momentum are precisely all future-pointing unit timelike vectors. These form a hypersurface P in the tangent bundle T M called the mass shell. The distribution function f , which represents the density of particles with given spacetime position and four-momentum, is a non-negative real-valued function on P . A basic postulate in general relativity is that a free particle travels along a geodesic. Consider a future-directed timelike geodesic parametrized by proper time.

Then its tangent vector at any time is future-pointing unit timelike. Thus this geodesic has a natural lift to a curve on P , by taking its position and tangent vector together.

This defines a flow on P . Denote the vector field which generates this flow by X. (This vector field is what is sometimes called the geodesic spray in the mathematics literature.) The condition that f represents the distribution of a collection of particles moving freely in the given spacetime is that it should be constant along the flow, i.e. that Xf = 0. This equation is the Vlasov equation, sometimes also known as the Liouville or collisionless Boltzmann equation.

To get an explicit expression for the Vlasov equation, it is necessary to introduce local coordinates on the mass shell. In the following local coordinates x^α on spacetime are always chosen such that the hypersurfaces x⁰=const. are spacelike. (Greek and Roman indices take the values 0, 1, 2, 3 and 1, 2, 3 respectively.) Intuitively this means that x⁰, which may also be denoted by t, is a time coordinate and that the x^a are spatial coordinates. A timelike vector is future-pointing if and only if its zero component in a coordinate system of this type is positive. It is not assumed that the vector ∂/∂x⁰ is timelike. One way of defining local coordinates on P is to take the spacetime coordinates x^αtogether with the spatial components p^a of the four-momentum in these coordinates.

Then the explicit form of the Vlasov equation is:

∂f /∂t + (p^a/p⁰)∂f /∂x^a− (Γ^a_βγp^βp^γ/p⁰)∂f /∂p^a = 0 (1.1) where Γ^α_βγare the Christoffel symbols associated to the metric gαβ. Here it is understood that p⁰is to be expressed in terms of p^aand the metric using the relation g_αβp^αp^β = −1.

An alternative way of coordinatizing the mass shell which is often useful is to use the components of the four-momentum in an orthormal frame, which has a priori nothing to do with the frame defined by the coordinates. It should be chosen so that the first vector is future-pointing timelike. Here only the case where the first vector is the unit normal to the hypersurfaces of constant time will be considered. The explicit form of the Vlasov equation in these coordinates is similar to (1.1), with the Christoffel symbols replaced by the Ricci rotation coefficients γ_νρ^µ of the frame. Explicitly, it is given by:

e⁰₀∂f /∂t + (v^µ/v⁰)e^a_µ∂f /∂x^a− (γⁱ_µνv^µv^ν/v⁰)∂f /∂vⁱ= 0 (1.2) The convention is used that frame indices are denoted by letters from the middle of the alphabet, while coordinate indices are taken from the beginning of the alphabet. The components of the frame vectors are denoted by e^α_µ and p^α and v^µ are related by the equation p^α= e^α_µv^µ. Since the frame is orthonormal v⁰=p1 + δ^ijvivj, where δ^ij is the Kronecker delta.

The Vlasov equation can be coupled to the Einstein equations as follows, giving rise to the Einstein–Vlasov system. The unknowns are a 4-manifold M , a (time orientable) Lorentz metric gαβ on M and a non-negative real-valued function f on the mass shell defined by gαβ. The field equations consist of the Vlasov equation defined by the metric gαβ for f and the Einstein equation Gαβ = 8πTαβ. (Units are chosen here so that the speed of light and the gravitational constant both have the numerical value unity.) To obtain a complete system of equations it remains to define Tαβ in terms of f and gαβ. It is defined as an integral over the part of the mass shell over a given spacetime point with respect to a measure which will now be defined. The metric at a given point of spacetime defines in a tautological way a metric on the tangent space at that point. The part of the mass shell over that point is a submanifold of the tangent space and as such has an induced metric, which is Riemannian. The associated measure is the one which we are seeking. It is evidently invariant under Lorentz transformations of the tangent space, a fact which may be used to simplify computations in concrete situations. In the coordinates (x^α, p^a) on P the explicit form of the energy-momentum tensor is:

Tαβ= − Z

f pαpβ|g|^1/2/p0dp¹dp²dp³ (1.3) A simple computation in normal coordinates based at a given point shows that Tαβ

defined by (1.3) is divergence-free, independently of the Einstein equations being satisfied.

This is of course a necessary compatibility condition in order for the Einstein–Vlasov system to be a reasonable set of equations. Another important quantity is the particle current density, defined by:

N^α= − Z

f p^α|g|^1/2/p0dp¹dp²dp³ (1.4)

A computation in normal coordinates shows that ∇αN^α= 0. This equation is an expres- sion of the conservation of the number of particles. There are some inequalities which follow immediately from the definitions (1.3) and (1.4). Firstly N_αV^α≤ 0 for any future- pointing timelike or null vector V^α, with equality only if f = 0 at the given point. Hence unless there are no particles at some point, the vector N^α is future-pointing timelike.

Next, if V^α and W^α are any two future-pointing timelike vectors then TαβV^αW^β ≥ 0.

This is the dominant energy condition ([12], p. 91). Finally, if X^α is a spacelike vector then TαβX^αX^β ≥ 0. This is the non-negative pressures condition. This condition, the dominant energy condition and the Einstein equations together imply that the Ricci ten- sor satisfies the inequality R_αβV^αV^β≥ 0 for any timelike vector V^α. The last inequality is called the strong energy condition. These inequalities constitute one of the reasons which mean that the Vlasov equation defines a well-behaved matter model in general relativity. However this is not the only reason. A perfect fluid with a reasonable equation of state or matter described by the Boltzmann equation also have energy-momentum tensors which satisfy these inequalities.

The Vlasov equation in a fixed spacetime is a linear hyperbolic equation for a scalar function and hence solving it is equivalent to solving the equations for its characteristics.

In coordinate components these are:

dX^a/ds = P^a

dP^a/ds = −Γ^a_βγP^βP^γ/P⁰ (1.5) Let X^a(s, x^α, p^a), P^a(s, x^α, p^a) be the unique solution of (1.5) with initial conditions X^a(t, x^α, p^a) = x^a and P^a(t, x^α, p^a) = p^a. Then the solution of the Vlasov equation can be written as:

f (x^α, p^a) = f₀(X^a(0, x^α, p^a), P^a(0, x^α, p^a)) (1.6) where f₀is the restriction of f to the hypersurface t = 0. This function f₀serves as initial datum for the Vlasov equation. It follows immediately from this that if f0 is bounded by some constant C, the same is true of f . This obvious but important property of the solutions of the Vlasov equation is used frequently without comment in the study of this equation.

The above calculations involving Tαβ and N^α were only formal. In order that they have a precise meaning it is necessary to impose some fall-off in the momentum variables on f so that the integrals occurring exist. The simplest condition to impose is that f has compact support for each fixed t. This property holds if the initial datum f0 has compact support and if each hypersurface t = t₀ is a Cauchy hypersurface. For by the definition of a Cauchy hypersurface, each timelike curve which starts at t = 0 hits the hypersurface t = t₀ at a unique point. Hence the geodesic flow defines a continuous mapping from the part of the mass shell over the initial hypersurface t = 0 to the part over the hypersurface t = t₀. The support of f (t₀), the restriction of f to the hypersurface t = t0is the image of the support of f0under this continuous mapping and so is compact. Let P (t) be the supremum of the values of |p^a| attained on the support of f (t). It turns out that in many cases controlling the solution of the Vlasov equation coupled to some field equation in the case of compactly supported initial data for the

distribution function can be reduced to obtaining a bound for P (t). An example of this is given below.

The data in the Cauchy problem for the Einstein equations coupled to any matter source consist of the induced metric g_ab on the initial hypersurface, the second funda- mental form kabof this hypersurface and some matter data. In fact these objects should be thought of as objects on an abstract 3-dimensional manifold S. Thus the data con- sist of a Riemannian metric g_ab, a symmetric tensor k_ab and appropriate matter data, all defined intrinsically on S. The nature of the initial data for the matter will now be examined in the case of the Einstein–Vlasov system. It is not quite obvious what to do, since the distribution function f is defined on the mass shell and so the obvious choice of initial data, namely the restriction of f to the initial hypersurface, is not appropriate.

For it is defined on the part of the mass shell over the initial hypersurface and this is not intrinsic to S. This difficulty can be overcome as follows. Let φ be the mapping which sends a point of the mass shell over the initial hypersurface to its orthogonal projection onto the tangent space to the initial hypersurface. The map φ is a diffeomorphism. The abstract initial datum f0for f is taken to be a function on the tangent bundle of S. The initial condition imposed is that the restriction of f to the part of the mass shell over the initial hypersurface should be equal to f₀ composed with φ. An initial data set for the Einstein equations must satisfy the constraints and in order that the definition of an abstract initial data set for the Einstein equations be adequate it is necessary that the constraints be expressible purely in terms of the abstract initial data. The constraint equations are:

R − k_abk^ab+ (trk)²= 16πρ (1.7)

∇ak^a_b − ∇b(trk) = 8πjb (1.8) Here R denotes the scalar curvature of the metric gab. If n^α denotes the future-pointing unit normal vector to the initial hypersurface and h^αβ = g^αβ+ n^αn^β is the orthogonal projection onto the tangent space to the initial hypersurface then ρ = T_αβn^αn^β and j^α = −h^αβTβγn^γ. The vector j^α satisfies j^αnα = 0 and so can be naturally identified with a vector intrinsic to the initial hypersurface, denoted here by j^a. What needs to be done is to express ρ and ja in terms of the intrinsic initial data. They are given by the following expressions:

ρ = Z

f0(p^a)p^apa/(1 + p^apa)^1/2(⁽³⁾g)^1/2dp¹dp²dp³ (1.9) ja=

Z

f0(p^a)pa(⁽³⁾g)^1/2dp¹dp²dp³ (1.10) If a three-dimensional manifold on which an initial data set for the Einstein–Vlasov system is defined is mapped into a spacetime by an embedding ψ then the embedding is said to induce the given initial data on S if the induced metric and second fundamental form of ψ(S) coincide with the results of transporting gaband kabwith ψ and the relation f = f₀◦ φ holds, as above. A form of the local existence and uniqueness theorem can now be stated. This will only be done for the case of smooth (i.e. infinitely differentiable) initial data although versions of the theorem exist for data of finite differentiability.

Theorem 1.1. Let S be a 3-dimensional manifold , gab a smooth Riemannian metric on S, kab a smooth symmetric tensor on S and f0 a smooth non-negative function of compact support on the tangent bundle T S of S. Suppose further that these objects satisfy the constraint equations (1.7 )-(1.8 ). Then there exists a smooth spacetime (M, gαβ), a smooth distribution function f on the mass shell of this spacetime and a smooth embedding ψ of S into M which induces the given initial data on S such that g_αβ and f satisfy the Einstein–Vlasov system and ψ(S) is a Cauchy hypersurface. Moreover , given any other spacetime (M⁰, g_αβ⁰ ), distribution function f⁰ and embedding ψ⁰ satisfying these conditions, there exists a diffeomorphism χ from an open neighbourhood of ψ(S) in M to an open neighbourhood of ψ⁰(S) in M⁰ which satisfies χ ◦ ψ = ψ⁰ and carries g_αβ and f to g_αβ⁰ and f⁰ respectively.

The formal statement of this theorem is rather complicated, but its essential meaning is as follows. Given an initial data set (satisfying the constraints) there exists a corre- sponding solution of the Einstein–Vlasov system and this solution is locally unique up to diffeomorphism. There also exists a global uniqueness statement which uses the notion of the maximal Cauchy development of an initial data set, but this is not required in the following. The first proof of a theorem of this kind for the Einstein–Vlasov system is due to Choquet-Bruhat [7].

In the following we are mainly concerned with asymptotically flat spacetimes. These are the spacetimes which are appropriate for describing isolated systems in general relativ- ity. It is assumed that these spacetimes admit a Cauchy hypersurface with topology R³, although more general cases could also be considered. The smooth data set (gab, kab, f0) on R³is said to be asymptotically flat if there exist global coordinates x^a such that as |x|

tends to infinity the components gabin these coordinates tend to δab, the components kab

tend to zero, f0 has compact support and certain norms are finite. These are weighted Sobolev norms. If u is a smooth function on R³ define

kukH^s

δ =

" _s X

i=0

Z

(1 + |x|²)^(δ+i)|Dⁱu|²dx

#1/2

(1.11) where |Dⁱu| denotes the maximum modulus of any partial derivative of order i of the function u. If the quantity (1.11) is finite then u is said to belong to the weighted Sobolev space H_δ^s. Assume for asymptotic flatness that g_ab− δab ∈ H_δ^s and that k_ab ∈ H_δ+1^s−1 for s sufficiently large and −3/2 < δ < −1/2. If a spacetime is asymptotically flat then Theorem 1.1 can be sharpened to say that there exists a local solution corresponding to the given initial data and coordinates defined for that solution such that the solution exists on R³×[0, T ) for some T > 0 and the data induced on the hypersurfaces t =const. satisfy the same type of asymptotic flatness conditions as the initial data on the hypersurface t = 0. More precisely, there exist coordinates so that the data induced by the solution on each hypersurface of constant time belongs to the same weighted Sobolev space as the initial data. Moreover the restrictions of the functions g₀₀+ 1 and g_0a of the metric components to any hypersurface of constant time belong to weighted Sobolev spaces. A proof of this is sketched in Section 4.

The local existence theorem can be supplemented by a statement that the solution depends continuously on the data. This is not stated in full generality here; only some

statements for asymptotically flat data are given. Given a family of initial data which is bounded in a weighted Sobolev space with s sufficiently large and is such that the metric is uniformly positive definite, the coordinates above can be chosen so that the solutions corresponding to all data in the family exist on the same time interval [0, T ), the weighted Sobolev norm of the data induced by these solutions on a hypersurface of constant time is uniformly bounded and the induced metric on one of these hypersurfaces is uniformly positive definite. Moreover, the restrictions of g₀₀+ 1 and g_0a to each hypersurface of constant time are uniformly bounded in a weighted Sobolev space for all data in the family and g00is uniformly bounded away from zero. See Section 4.

Suppose now that an asymptotically flat initial data set admits a group G of sym- metries, i.e. that a Lie group G acts on the manifold S in such a way that gab, kab and f0 are preserved. Then the spacetime in Theorem 1.1 can be chosen so that it admits G as a symmetry group. More precisely, there exists an action of G on M by isometries which preserves f and which restricts to the original action on S. To prove this, consider the geodesic γ(p) through a point p ∈ S orthogonal to S in a spacetime with the given initial data. Let t(p) denote the largest number such that, when γ(p) is parametrized by proper time, with p corresponding to the parameter value zero, this geodesic is defined on the interval (−t(p), t(p)). By what has been said above, the spacetime can be chosen so that there exists a number T > 0 which is smaller than t(p) for each p ∈ S. Since the spacetime is globally hyperbolic, each point can be joined to the initial hypersurface by a timelike geodesic of maximal length. If T is chosen sufficiently small then the geodesic is unique. A new spacetime can be defined as the open subset of the original spacetime where this distance is less than T . It will now be shown that the action of G on S extends to an action on this new spacetime, whose underlying manifold will be denoted by M . Given a point q ∈ M , let p be the point where the unique geodesic γ of maximal length from q to the initial hypersurface S meets S. Let φ : G×S → S denote the action of G on S. For g ∈ G and q ∈ M , let ˜φg(q) be the point which lies the same distance to the future of φ_g(p) along the future-directed geodesic starting orthogonal to S at φ_g(p) as q lies to the future of p along γ(p). This defines a mapping ˜φ : G × M → M by ˜φ(g, q) = ˜φg(q).

This mapping ˜φ is an action of G on M which restricts to φ. By the uniqueness part of Theorem 1.1, it must preserve gαβ and f .

The global theorems to be proved later make use of the concept of maximal hypersur- faces. A spacelike hypersurface in a spacetime is called maximal if its mean curvature trk is zero. If an initial data set is given which is maximal in this sense and asymptotically flat it is of interest to know whether the corresponding local solution of the Einstein–Vlasov system, whose existence is guaranteed by the above theorem, can be foliated by maximal hypersurfaces in a neighbourhood of the initial hypersurface. Given what has already been said about the local existence of asymptotically flat spacetimes, this can be proved using the implicit function theorem (cf. [16]). The time coordinate t above can be cho- sen so that its level hypersurfaces are maximal hypersurfaces and it can also be arranged that as |x| → ∞ the coordinate t agrees asymptotically with proper time along a geodesic which starts normal to the initial hypersurface. (This implies that g₀₀tends to unity as |x|

tends to infinity.) When this has been imposed the foliation by maximal hypersurfaces is unique and so this construction gives a unique preferred time coordinate. The restriction

of the solution to any of the maximal hypersurfaces is asymptotically flat. Once again there is a statement of uniformity. For a family of maximal initial data which is bounded in a suitable weighted Sobolev space the maximal foliation can be assumed to exist on a time interval which is uniform for all data in the family. These statements about the existence of maximal foliations are not too dependent on the fact that the matter is de- scribed by the Vlasov equation. The essential property which is needed for the existence and uniqueness theorems is the strong energy condition.

The approach to studying the global structure of asymptotically flat solutions of the Einstein–Vlasov system presented in these lectures is closely related to work which has been done in the spatially compact case [25, 4]. The main difference is that in a non-flat spacetime satisfying the strong energy condition with a compact Cauchy hypersurface there exists at most one maximal hypersurface [16]. In this case the maximal foliation of the asymptotically flat case can be replaced by a constant mean curvature (CMC) foliation. This means that each leaf of the foliation has constant mean curvature, while the value of the mean curvature varies monotonically from one leaf to the next.

2. Spherical symmetry. Investigating the global properties of general solutions of the Einstein–Vlasov system is beyond the scope of existing mathematical techniques.

For comparison, note that the same comment applies to the Vlasov–Maxwell system (cf.

[19] for the most general known results) while general global existence results have been obtained for the simpler Vlasov–Poisson system ([17], [13], [21]). When a system of partial differential equations appears inaccessible to direct attack, a natural strategy is to study the simpler equations obtained by imposing symmetry conditions on the solutions of the original equations. In the case of asymptotically flat solutions of the Einstein–Vlasov system, it seems that there are only two possible symmetry assumptions: spherical and rotational symmetry. In the latter case, where the symmetry group is one-dimensional and has fixed points (on the axis of rotation), the simplification obtained is not sufficient to bring the problem within range of present techniques. Thus in the following treatment of global questions we consider only the spherically symmetric case. Note for comparison that in the spatially compact case a wider variety of tractable symmetry types exists.

A solution (M, g_αβ, f ) of the Einstein–Vlasov system is said to be spherically sym- metric if there exists an action of SO(3) on M by isometries whose generic orbits are two-dimensional such that the natural lift of this action to the mass shell preserves f . There is of course an analogous definition of a spherically symmetric initial data set.

Consider now a spherically symmetric asymptotically flat maximal initial data set. From the last section we know that there exists a corresponding local solution of the Einstein–

Vlasov system. Moreover, it can be assumed without loss of generality that SO(3) acts on this local solution as a symmetry group so that it is spherically symmetric. Furthermore, there exists a neighbourhood U of the initial hypersurface which can be foliated by max- imal hypersurfaces whose intrinsic geometry is asymptotically flat and this foliation is unique. It follows from the latter fact that each maximal hypersurface is invariant under the action of SO(3). In other words, it is a union of orbits of the action of SO(3) on spacetime. Let r, the area radius, be defined by the condition that on any orbit it takes the constant valuepA/4π, where A is the area of the given orbit. Consider now a fixed

spacelike hypersurface S0which is invariant under the action of SO(3). A geodesic of the induced metric on S0 which starts orthogonal to the orbits remains orthogonal to them.

These geodesics will be called radial geodesics. Let θ, φ be standard spherical coordinates on one of the orbits. Extend them to be constant along the radial geodesics. Since radial geodesics can never intersect except at the centre this prescription is globally well de- fined. In the coordinates (r, θ, φ) the metric components g12 and g13vanish. The metric intrinsic to the orbits takes the standard form r²(dθ²+ sin²θdφ²). Hence, provided the gradient of r does not vanish anywhere, the induced metric on S0 can be written in the form

e^2λ(r)dr²+ r²(dθ²+ sin²θdφ²) (2.1) This way of defining spatial coordinates was used in [20]. It cannot be used if the restric- tion of r to the hypersurfaces of constant time has a vanishing gradient somewhere. A point where the gradient vanishes corresponds to a minimal surface. Here we use another type of coordinate system which does not suffer from this difficulty. First note that it is possible to write the metric in the form

B²(x)dx²+ r²(x)(dθ²+ sin²θdφ²) (2.2) without restriction, for some function B. Since there is always a neighbourhood of the origin without minimal surfaces it can be assumed without loss of generality that r(x) = x near the origin. Since B is smooth when considered as a function on spacetime, it must be a smooth function of x². Furthermore, in order that the spacetime be regular at x = 0 and not have a conical singularity, B(0) must be equal to unity. This means in particular that it is possible to write B(x) = 1 + x²D(x), where D(x) is a smooth function of x². A new coordinate R will be sought which is a function of x and has the property that, when expressed in terms of the coordinate R, the metric takes the form:

A²(R)(dR²+ R²(dθ²+ sin²θdφ²)) (2.3) Coordinates of this sort, or more precisely the Cartesian coordinates corresponding to these polar coordinates, are often known as isotropic coordinates. Writing down the coor- dinate transformation shows that R(x) is a solution of the equation dR/dx = RB(x)/r(x).

This equation has a solution which is unique up to a constant scaling. To see this, note first that this is a first order homogeneous linear ordinary differential equation. Hence it has a one-parameter family of solutions which can all be got by multiplying one par- ticular solution by an arbitrary constant. If we know the existence of the solution near the origin then the existence for all values of R follows by the standard existence and uniqueness theorem for ordinary differential equations. Near the origin the equation can be solved, giving R = Cx expRx

0 x⁰D(x⁰)dx⁰. Near infinity, the asymptotic flatness of the metric implies that A tends to a constant value. The scaling can be chosen so that limR→∞A(R) = 1 and then the solution is determined uniquely. Now let t be a time coordinate which is constant on each leaf of the preferred foliation by maximal hypersur- faces and which agrees asymptotically with proper time. Introducing a coordinate R as above on each leaf puts the metric in the form:

−α²(t, R)dt²+ A²(t, R)[(dR + β(t, R)dt)²+ R²(dθ²+ sin²θdφ²)] (2.4)

As a consequence of the choice of coordinates the functions A and α tend to unity as R → ∞ for each fixed t while β → 0. The smoothness of the spacetime metric together with spherical symmetry implies that α, A, and R⁻¹β are smooth functions of R².

For a metric of the form (2.4) which satisfies the extra condition that the hypersurfaces of constant time are maximal hypersurfaces the field equations and coordinate conditions take the following form:

(R²(A^1/2)⁰)⁰= −¹₈A^5/2R²(³₂K²+ 16πρ) (2.5) α⁰⁰+ 2α⁰/R + A⁻¹A⁰α⁰= αA²[³₂K²+ 4π(ρ + trS)] (2.6)

K⁰+ 3(A⁻¹A⁰+ 1/R)K = 8πAj (2.7)

β⁰− R⁻¹β = ³₂αK (2.8)

∂_tA = −αKA + (βA)⁰ (2.9)

∂tK = −A⁻²α⁰⁰+ A⁻³A⁰α⁰+ α[−2A⁻³A⁰⁰+ 2A⁻⁴A⁰²− 2A⁻³A⁰/R − 8πSR

+ 4πtrS − 4πρ] + βK⁰ (2.10)

The notation used in these equations will now be explained. A prime denotes a deriva- tive with respect to R. The quantity K is that obtained by contracting the second fundamental form of the hypersurface t =const. twice with the unit vector A⁻¹∂/∂R while SR is obtained in the corresponding way from the energy-momentum tensor. The quantity trS is the trace of the spatial part of the energy-momentum tensor, i.e. if n^α is the unit future-pointing normal vector to the hypersurfaces of constant time, then trS = Tαβ(g^αβ+ n^αn^β). The quantity j is obtained by contracting Tαβonce with n^αand once with the vector A⁻¹∂/∂R and ρ is the energy density Tαβn^αn^β. In the standard terminology of the 3 + 1-decomposition of Einstein’s equations, α is the lapse function and β is the one non-trivial component of the shift vector. Equation (2.5) is the ex- plicit form of the Hamiltonian constraint (1.7) in this class of spacetimes with this kind of coordinate condition. The maximal slicing condition is expressed by the lapse equa- tion (2.6). The one non-trivial component of the momentum constraint (1.8) in these spacetimes is (2.7). Equation (2.8) is a consequence of the coordinate condition chosen while (2.9) follows from the definition of the second fundamental form. Finally, (2.10) is the one non-trivial Einstein evolution equation in this class of spacetimes. This form of the field equations has been used by Shapiro and Teukolsky for numerical calculations (see [27]).

What has been shown above implies that given asymptotically flat spherically symmet- ric maximal initial data for the Einstein–Vlasov system, there exists a corresponding local solution and some T1> 0 such that this spacetime can be covered by coordinates which cast it in the form (2.4) and for which the time coordinate ranges in the interval (−T1, T1) and the initial hypersurface is given by t = 0. In fact we are only interested in evolution to the future and hence only consider the part of spacetime on the interval [0, T1). The quan- tities describing the metric and matter then satisfy the equations (2.5)-(2.10). The Vlasov equation is of course also satisfied. It turns out to be useful for some purposes to write it in Cartesian coordinates. Let x^a be the coordinates (R sin θ cos φ, R sin θ sin φ, R cos θ).

Define a related orthonormal frame by ei= A⁻¹∂/∂xⁱ. Then if the mass shell is coordina- tized by (t, x^a, vⁱ) , where vⁱ denote the components of a vector in the given orthonormal

frame the Vlasov equation takes the explicit form:

∂f

∂t + αA⁻¹ v

p1 + |v|² − βx R

!

·∂f

∂x +



−A⁻¹α⁰p

1 + |v|²x R −1

2αK v − 3vr

x R



−αA⁻²A⁰

vvr− |v|²x R

 1

p1 + |v|²

#

· ∂f

∂v = 0 (2.11)

Here a dot denotes the usual inner product in R³, |v| =√

v · v and vr= (v · x)/R. For given initial data there exists a solution on an interval [0, T1) of the equations (2.5)-(2.11) supplemented by the definitions of the matter quantities. The whole system of equations will be referred to as the ‘reduced Einstein–Vlasov system’. This solution of the reduced system is uniquely determined by its restriction to t = 0, as follows easily from the general uniqueness theorem for solutions of the Einstein–Vlasov system and the uniqueness of the coordinate system used. As a consequence there exists a greatest value of T₁ (finite or infinite, call it T∗) for which a solution of the reduced equations with the given initial data exists on the time interval [0, T1). The interval [0, T_∗) is called the maximal interval of existence. The global existence question, which is the main theme of these lectures, is the question under what circumstances T_∗= ∞.

One possible strategy for proving global existence theorems will now be outlined.

Suppose that in some way it were possible to show for given initial data that for any corresponding solution on a finite interval [0, T1) the metric components, the distribution function and all their derivatives of all orders with respect to t and x were bounded.

Then global existence for these initial data would follow. For the metric components, the distribution function and all their derivatives of all orders would be uniformly continuous.

By a standard theorem on metric spaces they would all extend to continuous functions on the closed interval [0, T1]. By another standard theorem, this time of real analysis, the extensions are C^∞and each derivative of each extension is equal to the extension of the corresponding derivative. In this way smooth initial data are defined on the hypersurface t = T1. Provided these new initial data are asymptotically flat, the local existence theorem can be applied again to show that the original solution has an extension to an interval [0, T₂) with T₂> T₁. Hence T₁6= T∗. But since T₁ was an arbitrary positive number this only leaves the possibility that T_∗= ∞ and global existence is proved. In the following a situation is exhibited where bounds similar to those which are assumed in this argument can actually be obtained.

To get closer to the situation which has just been described, consider a solution of the reduced equations defined on some interval [0, T₁). It will now be shown that many quantities can be bounded by using the Einstein–Vlasov system. By the dominant energy condition ρ ≥ 0. Hence equation (2.5) shows that R²(A^1/2)⁰ is a non-increasing function of R for each fixed t. When R = 0 it is zero and hence R²(A^1/2)⁰ ≤ 0. It follows that A⁰ ≤ 0. The boundary condition that A → 1 as R → ∞ then gives A ≥ 1. Next, from (2.6),

(R²Aα⁰)⁰ = αA³R²[³₂K²+ 4π(ρ + trS)] ≥ 0 (2.12) The expression on the right hand side of (2.12) is non-negative, as follows from the dominant energy and non-negative pressures conditions. Since R²Aα⁰vanishes for R = 0

it can be seen that R²Aα⁰ ≥ 0 and α⁰≥ 0. Using the boundary condition that α → 1 as R → ∞ gives α ≤ 1.

Next an estimate of Malec and ´O Murchadha [15] will be used. The expansions of the null geodesics which start normal to the orbits are given by

θ = 2(A⁻²A⁰+ (AR)⁻¹) + K θ⁰= 2(A⁻²A⁰+ (AR)⁻¹) − K (2.13) The area radius is given by r = AR. The theorem of [15] states that, if the dominant energy condition holds, the quantities rθ and rθ⁰are bounded in modulus by two. Adding and subtracting these estimates and using the explicit expressions in (2.13) gives the inequalities RA|K| ≤ 2 and |RA⁻¹A⁰+ 1| ≤ 1. Since A ≥ 1 it can be concluded from the first of these inequalities that |K| ≤ 2R⁻¹. The second inequality gives |A⁻¹A⁰| ≤ 2R⁻¹. In particular this gives pointwise bounds for K and A⁻¹A⁰ away from the centre.

Integrating (2.8) gives

R⁻¹₂ β(R2) − R⁻¹₁ β(R1) =³₂ Z R₂

R₁

(αK/s)ds (2.14)

Asymptotic flatness implies that K = O(R⁻¹) as R → ∞. Hence it is possible to let R2

tend to infinity in this equation to get an expression for R⁻¹β(R) as an integral from R to ∞. Using the bounds already obtained for K and α shows that

|β| ≤ ³₂R Z ∞

R

(2/s²)ds ≤ 3 (2.15)

and so β is bounded. A bound for α⁰ can be obtained by analysing the inequality (R²Aα⁰)⁰ ≥ 0, which was already used to show that α ≤ 1. Integrating this between the radii R₁ and R₂ with R₁< R₂ gives:

α⁰(R2) ≥ (R1/R2)²(A(R1)/A(R2))α⁰(R1) (2.16) Integrating again then gives:

α(R2) ≥ α⁰(R1)(R²₁A(R1)) Z R₂

R1

R⁻²(A(R))⁻¹dR (2.17) Now use the facts that α(R2) ≤ 1 and that A(R) ≤ A(R1) for R ≥ R1 to see that:

1 ≥ α⁰(R₁)R₁² Z R₂

R₁

R⁻²dR (2.18)

This holds for all R2 ≥ R1 and so it is permissible to replace the upper limit in the integral by infinity. Evaluating the integral gives α⁰(R) ≤ R⁻¹ for all R > 0. Note that in deriving all these estimates, the only properties of the matter fields used were the dominant energy condition and the inequality ρ + trS ≥ 0. The latter follows from the strong energy condition. Thus all these estimates hold not only for the Einstein–

Vlasov system, but also for the Einstein equations coupled to any matter model which satisfies the dominant and strong energy conditions. This includes perfect fluids with reasonable equations of state, matter described by the Boltzmann equation, the massless scalar field (or more generally wave maps), any of these matter models combined with an electromagnetic field in such a way that the total energy-momentum tensor is the sum of the individual energy-momentum tensors, and the Yang-Mills equations for any semi-simple gauge group.

These estimates give good information on the solution away from the centre but, except in the case of the estimate for β, give no control at the centre. Pointwise estimates for α⁰, A⁰ and K which also give useful information at the centre can be obtained by optimization arguments. For any fixed radius R₀ we have

|K(R)| ≤ 4πA⁴(0)kρk∞R0 (2.19)

for any R ≤ R0. Here k k_∞ denotes the L^∞norm in space, i.e. the maximum value of a function on a hypersurface of constant time. This inequality is obtained by integrating equation (2.7) and using the dominant energy condition to bound the modulus of j by ρ.

On the other hand, if R ≥ R0 then |K(R)| ≤ 2R⁻¹₀ . Thus for any value of R it is true that

|K(R)| ≤ 4πA⁴(0)kρk_∞R₀+ 2R⁻¹₀ (2.20) This can be optimized by choosing R₀ so that the function of R₀ occurring on the right hand side of this last inequality has a critical point. This occurs when R0 is equal to [2πA⁴(0)kρk_∞]^−1/2. It follows that kKk_∞≤ CA²(0)kρk^1/2∞ for some constant C. A similar procedure can be used to estimate α⁰.

α⁰(R) = A⁻¹R⁻² Z R

0

αA³s²(³₂K²+ 4πρ + 4πtrS)ds ≤ CA⁷(0)kρk_∞R₀ (2.21) for R ≤ R0. Combining this with the previous pointwise estimate for α⁰ gives

α⁰≤ C(A⁷(0)kρk_∞R0+ R₀⁻¹) (2.22) Doing an optimization as above leads to an estimate of the form α⁰ ≤ CA^7/2(0)kρk^1/2∞ . From the equation for A:

A⁰(R) =¹₄R⁻²A^1/2 Z R

0

A^5/2s²(³₂K²+ 16πρ)ds (2.23) Thus

|A⁰| ≤ CA(0)[¹₄A⁶(0)kρk_∞R0+ 2R⁻¹₀ ] (2.24) Optimizing gives kA⁰k_∞≤ CA⁴(0)kρk^1/2∞ . Starting from the same equations we can also bound R⁻¹K, R⁻¹α⁰ and R⁻¹A⁰ pointwise in terms of kρk_∞ and A(0). In particular R⁻¹K can be bounded by a constant multiple of A⁴(0)kρk. Equations (2.6) and (2.7) then allow α⁰⁰ and K⁰ to be bounded. Solving (2.5) for A⁰⁰shows that it too can be bounded.

Equation (2.14) and the bounds for β and R⁻¹K imply that R⁻¹β can be bounded by a constant multiple of A²(0)kρk^1/2∞ . Using equation (2.8) then gives a similar bound for β⁰. In the proof of variants of these estimates discussed in the next section, the con- servation of the total (ADM) mass plays a role and it is convenient to say something about this conservation law at this point. It is a quite general property of asymptotically flat spacetimes. It is particularly easy to see in the situation considered here, where outside a compact set the spacetime is vacuum and spherically symmetric. Equation (2.7) can be rearranged to give (A³R³K)⁰= 8πR³A⁴j. In vacuum this integrates to give K = K₀(t)R⁻³A⁻³ for some function K₀(t). Putting this in (2.5) and using the vacuum condition again gives (R²(A^1/2)⁰)⁰= O(R⁻⁴). This can be integrated to give:

A(t, R) = (1 + A0(t)R⁻¹)²+ O(R⁻⁴) (2.25)

It follows from (2.8) that (R⁻¹β)⁰ = ³₂R⁻¹αK = O(R⁻⁴). Hence β = O(R⁻²) and β⁰ = O(R⁻³). It then follows from (2.9) that ∂tA = O(R⁻³). Integrating this last equation in time from 0 to t shows that A(t, R) = A(0, R) + O(R⁻³), so that A0(t) is in fact independent of t. The ADM mass is given by lim_R→∞(−R²A⁰) and so is also time independent. From (2.5) we can calculate that the ADM mass is equal to

mADM = ¹₈ Z ∞

0

A^5/2R²(³₂K²+ 16πρ)dR (2.26) In all these estimates only the dominant and strong energy condtions have been used.

To go beyond the results of the last paragraphs it is necessary to use the specific nature of the matter model. For the Einstein–Vlasov system a continuation criterion will be proved. It is formulated in terms of a quantity P (t), which is defined to be the largest momentum of any particle at time t. In other words

P (t) = sup{|v| : f (t, x, v) 6= 0 for some x} (2.27) Before stating the continuation criterion it is necessary to take some time to discuss the relation between the differentiability of the functions of R describing the spacetime and the differentiability of the corresponding objects in spacetime. The simplest case is that of the scalar functions α and A. They are C^∞ in the spacetime sense if and only if they are C^∞ as functions of R and all the derivatives of odd order vanish at the origin. This follows from Lemma A1 of the appendix with m = 0. There is also a quantitative version of this, which follows from Lemma A2. Consider next β. By definition β = β^axa/R and β^a = βx^a/R, where β^a is the shift vector. Because of spherical symmetry β must vanish at the origin. Hence we can apply Lemma A3 to β^a. Lemma A4 gives quantitative results for β^a. Consider next K. By definition A²K = k_abx^ax^b/R². The maximal hypersurface condition and spherical symmetry together imply that kabvanishes at the origin. Hence Lemma A2 can be applied with m = 2. Similar considerations apply to the matter quantities j and S_R, whereby in the latter case it is necessary to write S_R= ˜S_R+¹₃trS, with ˜SR being the contribution to SR of the trace free part of Tab. Because of spherical symmetry ˜SR vanishes at R = 0. The following expressions for some of the quantities occurring in the Vlasov equation are also significant:

βxa/R = βa kab= −¹₂KA²(δab− 3xaxb/R²) α⁰x_a/R = ∇_aα

A⁰xa/R = ∇aA

(2.28)

Theorem 2.1. If a solution of the reduced Einstein–Vlasov system on the interval [0, T ) for some positive real number T is such that P (t) and A(t, 0) are bounded then the solution extends to an interval [0, T1) with T1> T . In particular , if the maximal interval of existence [0, T_∗) is finite then either P (t) or A(t, 0) is unbounded there.

P r o o f. Suppose that P (t) is bounded on the interval [0, T ). Then the matter quanti- ties ρ, trS and SRare bounded there. It has been shown above that this, together with the boundedness of A(t, 0), implies that the quantities A, A⁰, A⁰⁰, R⁻¹A⁰, α, α⁰, α⁰⁰, R⁻¹α⁰, K, K⁰, R⁻¹K. β, β⁰, and R⁻¹β are bounded. It remains to show that all higher spacetime derivatives of all these quantities are bounded. We have a C²bound for α and A and a

C¹ bound for K and β when these are considered as functions of R. These imply a C² bound for A, α and a C¹bound for β^a and kabin the three dimensional sense, using the results of the appendix. It follows that a C¹ bound for all the coefficients of the Vlasov equation on the support of f is obtained. The equations obtained by differentiating the Vlasov equation with respect to x and v then give the boundedness of the first derivatives of f with respect to x and v. Using the definition of the energy-momentum tensor gives a C¹ bound for its Cartesian components. The results of the appendix then imply a C¹ bound for ρ, j and trS.

Higher derivatives can now be bounded inductively. Assume that a solution of the reduced equations on a given time interval is such that the C^k+1norms of α and A and the C^k norms of K, β, f , ρ, j, trS and S_Rare bounded and that A⁻¹is also bounded. Note that it has already been shown that under the hypotheses of the theorem this statement holds for k = 1 and this suffices to start the induction. Now consider the case of general k. It is convenient to rewrite some of the reduced equations in the following form:

(A^1/2)⁰(R) = −¹₈R⁻² Z R

0

s²[A^5/2(³₂K²+ 16πρ)](s)ds (2.29)

A(R)α⁰(R) = R⁻² Z R

0

s²[A²(³₂K²+ 4π(ρ + trS))](s)ds (2.30)

A³(R)K(R) = R⁻³ Z R

0

s³[4πA⁴j](s)ds (2.31)

β⁰(R) = β⁰(0) + R Z R

0

s⁻¹[³₂αK](s)ds (2.32)

Applying Lemma A5 to (2.29) and (2.30) gives C^k+1bounds for (A^1/2)⁰and Aα⁰, consid- ered as functions of R. Combining this with the information already available gives C^k+2 bounds for A and α, considered as functions of R. In a similar way, (2.31) and Lemma A5 give a C^k+1bound for K. The quantity β⁰(0) is already known to be bounded. Hence (2.32) and Lemma A6 imply a C^k+1 bound for β. (In fact it implies a C^k+3 bound, but that is not relevant here.) Moreover, it can be checked that the derivatives of these functions which are required to vanish in order that the functions have the correspond- ing differentiability when considered as functions of three variables, according to the results of the appendix, do so. For the same reason bounds for the derivatives of these functions of three variables are obtained. It follows in particular that the coefficients of the Vlasov equation are C^k+1. Hence the solution of the Vlasov equation is bounded in the C^k+1 norm. An immediate consequence is that the Cartesian components of the energy-momentum tensor are bounded in the C^k+1 norm. Finally, applying the results of the appendix again shows that the C^k+1 norms of ρ, j and trS are bounded and this completes the inductive step.

It was mentioned earlier that in order to prove global existence it would suffice to bound the derivatives of all orders of all quantities of interest with respect to t and R.

Here only the derivatives with respect to R have been bounded and it turns out to be difficult to bound the time derivatives of the lapse function directly. Fortunately it is enough, in the present context, to bound the spatial derivatives, as will now be shown.

Let tn be a sequence of times with tn < T for each n and limn→∞tn = T . The initial data induced by the given solution on the hypersurfaces t = tn define a sequence of initial data which are bounded in the C^∞topology. In fact they are also bounded in the topology of a weighted Sobolev space. To prove this it suffices to obtain some estimates on an exterior region, say that defined by R ≥ 1. Equation (2.29) and the conservation of ADM mass shows that A⁰is O(R⁻²), uniformly in t. Equation (2.31) and the fact that the support of the matter is contained in a region of the form R ≤ R₀ for all t in the interval [0, T1) shows that K = O(R⁻³), uniformly in t. Using (2.25) and the fact that under the given circumstances the O(R⁻⁴) error term there is uniform in t shows that A − 1 = O(R⁻¹), uniformly in t. Thus gab− δab is bounded in H_δ¹ and kab is bounded in H_δ+1⁰ for −3/2 < δ < −1/2. To apply the more precise version of the local existence theorem for asymptotically flat spacetimes it is necessary to have a similar statement for weighted Sobolev spaces of higher order. This can be proved straightforwardly by induction using the equations (2.5) and (2.7). It follows that the solutions of the Einstein–

Vlasov system corresponding to the data on each of the hypersurfaces of constant t exist on some time interval of length  about the initial time where data are given, with  independent of n. Hence the solution extends to the interval [0, T + ).

With this result in hand, it is easy to show that the first singularity, if one exists, must occur in the centre.

Theorem 2.2. If a solution of the reduced Einstein–Vlasov system on the interval [0, T ) for some positive real number T is such that it has a smooth extension to an open neighbourhood of the point with coordinates (T, 0) then the solution extends to an interval [0, T₁) with T₁ > T . In particular , if the maximal interval of existence [0, T_∗) is finite then the solution has a singularity at the point (T∗, 0)

P r o o f. The neighbourhood occurring in the hypotheses of the theorem contains all points with t > T1 and R ≤ R1 for some T1 < T and some R1 > 0. Since the solution is smooth for t < T₁ it follows that all unknowns in the reduced system are bounded on the region R ≤ R₁, 0 ≤ t < T . The value of A at any point of a hypersurface of constant time can be bounded by its value at the centre at the given time and so A is bounded on the interval [0, T ). It was shown earlier that on any region of the form R ≥ R1 the quantities A⁻¹A⁰ K and α⁰are uniformly bounded. Hence under the present assumptions the quantities A⁰, K and α⁰ are bounded on the interval [0, T1). It was also shown that β is always bounded everywhere. It can be concluded that all the functions of t and x occurring as coefficients in the Vlasov equation are bounded. The characteristic system of the Vlasov equation then implies an inequality of the form P (t) ≤ P (0) + CRt

01 + P (s)ds.

By Gronwall’s lemma P (t) is bounded on the interval [0, T1), and applying Theorem 2.1 completes the proof.

R e m a r k. It is clear from the proof that the existence of an extension could be replaced by the assumption that there exists some R₁> 0 such that ρ and A are bounded on the region R ≤ R1.

Theorems 2.1 and 2.2 are analogues of Theorem 3.2 of [20] and Theorem 4.1 of [22]

respectively. It is instructive to compare the theorems involving maximal-isotropic coor-

dinates proved here with those involving Schwarzschild coordinates proved in [20] and [22]. The continuation criterion of Theorem 2.1 appears at first sight weaker than that of [20] since it is assumed that not only P (t) but also A(t, 0) is bounded. However, it is much easier to pass from Theorem 2.1 to Theorem 2.2 than it is to pass from the continuation criterion of [20] to the regularity theorem of [22]. Moreover, the passage from Theorem 2.1 to Theorem 2.2 does not involve any deep analysis of the Vlasov equation, which the proof of Theorem 4.1 of [22] does. Thus it is reasonable to hope that the method of proof used here can more easily be adapted to matter models other than the Vlasov equation than the approach using Schwarzschild coordinates. In the next section it will be seen that the apparently weaker continuation criterion given by Theorem 2.1 is also good enough to be applied in the proof of a global existence theorem for small initial data.

At this point some further remarks on the notion of ‘well-behaved’ matter models are in order. Consider the case of dust, i.e. a perfect fluid without pressure. In fact (see [24]), smooth solutions of the Einstein-dust equations can be considered as distributional solu- tions of the Einstein–Vlasov system. Dust satisfies the dominant energy and non-negative pressures conditions. However it cannot be expected that an analogue of Theorem 2.2 holds for dust. The reason is the occurrence of so-called shell-crossing singularities, which do not occur at the centre. As has been discussed in [24] and [23] this is of significance for the formulation of the cosmic censorship hypothesis and Theorem 2 can be taken as an indication that the Einstein–Vlasov system is a good starting point for studying the cosmic censorship hypothesis and has advantages over other, superficially simpler matter models, such as a perfect fluid. This is one of the main motivations for investigating the global properties of solutions of these differential equations.

3. Global existence for small data. In the last section a continuation criterion was given for solutions of the reduced equations. Now it will be applied to obtain a global existence theorem in a particular situation, namely that of small data. The notion of smallness of initial data is defined in the present context in terms of three quantities which characterize the size of the data. Let F0= kf (0)k_∞, P0= P (0) and let R0 be the smallest value of R such that f (0, R) vanishes for R > R₀.

Theorem 3.1. Let K be a fixed positive constant and consider initial data for the reduced equations with R0 ≤ K and P0 ≤ K. Then there exists an  > 0 such that for all data of this type which, in addition, satisfy F₀ <  the corresponding solution exists globally in time and the spacetime which it defines is timelike and null geodesically complete.

R e m a r k s 1. The spacetimes of the theorem are also spacelike geodesically complete but this will not be proved here. It is the completeness of timelike and null geodesics which is most interesting physically, since these represent the wordlines of particles.

2. The statement on geodesic completeness is an important part of the theorem since a theorem on global existence in some coordinate time does not necessarily imply any interesting invariant information.

In fact more detailed information concerning the asymptotic behaviour of the space- times covered by the theorem will be obtained. In particular, information will be obtained